Pol

Input: Pol()
$$x$$

A formal indeterminate (formal variable) which may be used to construct polynomials or rational functions as algebraic objects.

Input: Pol(n)
$$x_{n}$$

A formal indeterminate (formal variable) which may be used to construct polynomials or rational functions as algebraic objects, distinguished by an index $n$.

Input: Tuple(Pol(1), Pol(2), Pol(3))
$$\left(x_{1}, x_{2}, x_{3}\right)$$

Three formal indeterminates.

Input: And(NotEqual(Pol(1), 0), NotEqual(Pol(1), Pol(2)))
$$x_{1} \ne 0 \;\mathbin{\operatorname{and}}\; x_{1} \ne x_{2}$$

A formal indeterminate is not the same thing as an expression-level variable; it is a concrete mathematical object, and in particular an object distinct from any number and from any other indeterminate.

Input: And(Equal(Pol(n) - Pol(n), 0 * Pol(n), 0), Equal(Pol(n)**0, Pol(n) / Pol(n), 1))
$$x_{n} - x_{n} = 0 x_{n} = 0 \;\mathbin{\operatorname{and}}\; {x}_{n}^{0} = \frac{x_{n}}{x_{n}} = 1$$

Indeterminates of this type generates algebras with ordinary numbers as coefficients. They are not suitable for working in rings with different Zero and One elements.

Input: Equal(Pol(1) * Pol(2), Pol(2) * Pol(1))
$$x_{1} x_{2} = x_{2} x_{1}$$

Indeterminates of this type commutes with numbers and with other indeterminates of the same kind.

Input: Where(Equal(Sum(Pol(i), For(i, 1, m))**n, Sum((Factorial(n) / Product(Factorial(Item(k, i)), For(i, 1, m)) * Product(Pol(i)**Item(k, i), For(i, 1, m))), ForElement(k, S))), Def(S, Set(k, ForElement(k, CartesianPower(Range(0,n), m)), Equal(Sum(Item(k, i), For(i, 1, m)), n))))
$${\left(\sum_{i=1}^{m} x_{i}\right)}^{n} = \sum_{k \in S} \frac{n !}{\prod_{i=1}^{m} \left({k}_{i}\right)!} \prod_{i=1}^{m} {x}_{i}^{{k}_{i}}\; \text{ where } S = \left\{ k : k \in {\{0, 1, \ldots, n\}}^{m} \,\mathbin{\operatorname{and}}\, \sum_{i=1}^{m} {k}_{i} = n \right\}$$

The multinomial theorem for formal indeterminates.

Last updated: 2020-03-06 00:22:16